Constraining gamma-ray burst parameters with the first ultra-high energy neutrino event KM3-230213A

Context. The detection of the highest energy neutrino observed to date by KM3NeT, with an estimated energy of 220 PeV, opens up new possibilities for the study and identification of the astrophysical sources responsible for a di ffuse flux of such ultra-high-energy neutrinos, among which gamma-ray bursts are longstanding candidates.Aims. Based on the event KM3-230213A, we derived constraints on the baryon loading and density of the surrounding environment in models of blast waves in long-duration gamma-ray bursts.Methods. We computed the di ffuse flux from gamma-ray burst blast waves, either expanding in a constant density interstellar medium or developing in a radially decreasing density of a wind-like environment surrounding the gamma-ray burst progenitor star, by taking into account the expected neutrino spectra and luminosity function. We used a Poisson likelihood method to constrain the blast wave model parameters by calculating the expected number of neutrino events within the 90% confidence level energy range of KM3-230213A and by using the joint exposure of KM3NeT/ARCA, IceCube, and Pierre Auger.Results. We constrain the baryon loading to be f(b) <= 51 at 90% confidence, with the best-fit and 68% confidence interval being f(b) = 26.9(-17.2)(+11.4) for a constant interstellar medium particle density of n(0) = 1 cm(-3). In the wind-like environment case, the baryon loading is fb similar to 1095 at 90% confidence, with the corresponding 68% confidence interval being f(b) is an element of [8, 231], which is proportional to the sixth power of a variable density parameter of A(*) = 0.1.